by Fred Espenak, GSFC Planetary Systems Laboratory

The Lunar Limb Profile and Eclipse Predictions

Fred Espenak

Introduction

Earth's orbit around the Sun and the Moon's orbit around Earth are known to a very high precision.
Newcomb's theory of the motion of the Sun [1895] is accurate to within 0.5 arc-seconds for modern day eclipse predictions.
Brown's [1919] lunar theory which was later modified as the Improved Lunar Ephemeris [1954] has a similar precision.
Better yet, the Jet Propusion Laboratory's solar system ephemeris (known as DE405) is capable of Sun and Moon positions with errors of less than 0.1 arc-seconds.
This enables astronomers to predict the path of solar eclipses across Earth's surface with a positional accuracy of 0.4 kilometers or better.

For the present era, a far greater source of uncertainty arises in our knowledge of the Moon's surface topography (mountains and valleys) which are silhouetted in profile along the lunar limb during an eclipse.
Some of these features depart from a perfectly spherical Moon by about ± 3 arc-seconds.
This is nearly ± 6 kilometers (1 arc-second equals 1.863 km at the Moon's mean distance of 384,400 km).

The primary impact of the lunar limb profile is to introduce uncertainties of several seconds into the exact eclipse contact times and the duration of annularity or totality.

The Lunar Limb Profile

Solar eclipse contact times, the magnitude and the duration of totality (or annularity) all ultimately depend on the angular diameters and relative velocities of the Moon and Sun.
Unfortunately, these calculations are limited in accuracy by the departure of the Moon's limb from a perfectly circular figure.
The Moon's surface exhibits a rather dramatic topography, which manifests itself as an irregular limb when seen in profile.
Most eclipse calculations assume some mean lunar radius that averages high mountain peaks and low valleys along the Moon's rugged limb.
Such an approximation is acceptable for many applications, but if higher accuracy is needed, the Moon's actual limb profile must be considered.
Fortunately, an extensive body of knowledge exists on this subject in the form of Watts' limb charts [Watts, 1963].

These data are the product of a 1950's photographic survey conducted by C.B. Watts (U.S. Naval Observatory) in which he measured the heights of lunar features along the edge of the Moon as recorded in thousands of photographs taken at different phases and librations.
The limb profile heights are normalized with respect to an adopted smooth reference surface (or datum).
Analyses of lunar occultations of stars by Van Flandern [1970] and Morrison [1979] have shown that the average cross-section of Watts' datum is slightly elliptical rather than circular.
Furthermore, the implicit center of the datum (i.e. - the center of figure) is displaced from the Moon's center of mass.
In a follow-up analysis of 66000 occultations, Morrison and Appleby [1981] have found that the radius of the datum appears to vary with libration.
These variations produce systematic errors in Watts' original limb profile heights that attain 0.4 arc-seconds at some position angles.
Thus, corrections to Watts' limb profile data are necessary to ensure that the reference datum is a sphere with its center at the center of mass.

The Watts charts have been digitized by Her Majesty's Nautical Almanac Office in Herstmonceux, England, and transformed to grid-profile format at the U. S. Naval Observatory.
In this computer readable form, the Watts limb charts lend themselves to the generation of limb profiles for any lunar libration.
Ellipticity and libration corrections may be applied to refer the profile to the Moon's center of mass.
Such a profile can then be used to correct eclipse predictions which have been generated using a mean lunar limb.

The correction of eclipse predictions using Watts limb data results in agreement between predicted and observed contact times and durations to better than 0.5 seconds.
Without the corrections, the times and durations may be in error by as much as 2 to 3 seconds (and more near the path limits where the geometry is far more critical).
In most cases, such corrections are only necessary for scientific investigations where timings or limb profile effects are critical.

In recent years, better limb profile data has become available through the Kaguya and the Lunar Reconnaissance Orbiter (LRO) missions.
These data can be used to improve the precision of eclipse contact times and durations to the ~0.2 second level of accuracy.

References for Lunar Limb Profile

Brown, E. W., Tables of the Motion of the Moon, Yale University Press, New Haven, 1919.